Not all countries have legal reserve requirements. Yet, their banks still take deposits. See https://en.wikipedia.org/wiki/Reserve_requirement#Required_r... for an overview of the current situation. A look at history is also useful.

'Banks lend out deposits' is a useful approximation, and more true than its opposite.

To be more complete:

In the absence of legal requirement, yes, a bank could just create a deposit and a loan out of thin air. That's basically how all loans work in the first place:

When Alice gets a loan from ACME bank, roughly the following happens: the bank adds one million dollars to Alice's current account, and also adds an entry to their books that Alice owes them one million dollars.

So far so good, and no new deposits required, or any kind of funding at all, really.

If that was all that was happening, the bank would sit there and happily collect the interest differential between the loan and the current account.

And in this version of the world, you are right: the only thing keeping the bank from creating endless loans would be legal requirements.

Alas, our bank's life ain't so easy.

Alice won't just let the money he borrowed sit in her bank account. She will spend it.

Assume she spends it via a bank transfer to Bob who has an account with a Badger bank.

Unlike Alice or even Bob, Badger won't settle for just an entry in ACME's books. They demand reserves in interbank settling.

(Similar logic applies when Alice withdraws cash.)

The bank can get reserves from their own equity, or via loans (ie deposits, the interbank system, issuing bonds, etc).

Usually banks have a bit of a cushion, so on the margin they can make a few extra loans first, and look for extra funding afterwards.

This story from the perspective of a single bank does not change when a central bank messes with the amount of total reserves. The bank will still have to acquire reserves to fund lending.

(For the sake of our explanation, we need to differentiate between the central bank adding reserves via eg open market purchases of assets, and the central bank directly lending to banks.

For the former, really nothing changes from the bank's point of view: in order to sell a T-bill to the Fed, they first need to have a T-Bill already on their balance sheet and that assets needs to be funded by liability of either equity or loans, ie deposits in the wider sense.

For the latter, things get a bit more complicated from the bank's point of view.

However, from a perspective of the system as a whole, it doesn't matter too much how the Fed injects extra resources.

And yes, legal reserve requirements are sometimes a binding constraint. Then your original story is all there is to it. But that's a rather special and pathological situation.

For historic comparison, in the heyday of Scottish banking around the time of the Industrial Revolution, banks usually ran with about 2% reserves but about 30% capital. Neither of which was mandated by law, but emerged in competition.

Crucially, Scottish banks were allowed to print their own bank notes, so a customer withdrawing cash didn't affect a bank's reserves. Unless they asked for gold coins, which famously almost no-one ever did in Scotland at the time.

No one was forced to accept a bank's notes, but competitors readily took them at face value, and presented them in interbank settling to get reserves.)